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Statistical Inference Methods

Proportional

Confidence Interval

One Sample z-interval

Conditions:

  • Random/Representative Sample
  • Independent Samples
  • Normality: \(n\hat{p}\ge10\) and \(n(1-\hat{p})\ge10\)

$$ \hat{p} \pm z^\star\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

$$

Assuming the sample was selected in a reasonable manner, we are CLEVEL% confident that the actual proportion of VAR is between LOW and HIGH

Two Sample z-interval

  • Both random/representative samples
  • Both independent samples
  • Normality on both samples: \(n\hat{p}\ge10\) and \(n(1-\hat{p})\ge10\)
\[ (\hat{p}_1 - \hat{p}_2) \pm z^\star\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}} \]

Assuming the sample was selected in a reasonable manner, we are CLEVEL% confident that the difference of proportions of VAR is between LOW and HIGH

Hypothesis Tests

One Sample z-test

$$ h_0: p=p_0

\

h_a:p>, \ne, < p_0 $$

Conditions

  • Random/representative samples
  • Independent samples
  • Normality: \(n \ge 30\)
\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \]

Reject \(h_0\) when the p-value is less than \(\alpha\) (there is convincing evidence to support the alternative hypothesis), otherwise fail to reject (there lacks convincing evidence to support the alternative hypothesis).

Two Sample z-test

$$ h_0: p_1=p_2

\

h_a:p_1>, \ne, < p_2 $$

Conditions

  • Random/representative samples
  • Independent samples
  • Normality: \(n \ge 30\)

$$ \Large \hat{p_c}=\frac{n_1\hat{p_1}+n_2\hat{p_2}}{n_1+n_2} \\ z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\frac{\hat{p}_c(1-\hat{p}_c)}{n_1} + \frac{\hat{p}_c(1-\hat{p}_c)}{n_2}}}

$$

Assuming the sample was selected in a reasonable manner, we are CLEVEL% confident that the difference of proportions of VAR is between LOW and HIGH

Mean

Confidence Interval

One Sample t-interval

Conditions:

  • Random/representative sample
\[ \bar{x} \pm t^\star\frac{s}{\sqrt{n}} \]

Two Sample t-interval (dependent & paired)

$$ (\bar{x}_1 - \bar{x}_2) \pm t^\star \frac{s_d}{\sqrt{n}}

$$

Two Sample t-interval (independent)

$$ (\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}

$$

Hypothesis Test

One Sample t-test

\[ t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}} \]

Two Sample t-test (dependent)

\[ t^\star = \frac{\bar{x_d}}{\frac{s_d}{\sqrt{n}}} \]

Two Sample t-test (independent)

\[ t^\star = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]

Power of Tests

P-Hat: Sample Proportion

P-Naught: Hypothesized Proportion

P: Population Proportion

P-Vale: Probability that you would observe p-hat or something more extreme if p-naught were true

Power of a test: Probability of h0 rejection when it is false or should be rejected \(1-\beta\). Higher power indicates smaller beta. To increase power: Increase n (increases SE - curve gets skinner), or increase alpha (increases rejection zone)